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A MAXIMIZATION PROBLEM 39

Mathematical Statement of the GulfGolf Problem
Try Problem 13 to test The mathematical statement or mathematical formulation of the GulfGolf problem
your ability to formulate a is now complete. We succeeded in translating the objective and constraints of the
mathematical model for a
maximization linear problem into a set of mathematical relationships referred to as a mathematical
programming problem model. The complete mathematical model for the problem is then:
with less-than-or-equal-
to constraints. Max 10S þ 9D
subject to (s.t.)
7
/ 10 S þ 1D 630 Cutting and dyeing
1 / 2 S þ / 6 D 600 Sewing
5
2
1S þ / 3 D 708 Finishing
7 1
/ 10 S þ / 4 D 135 Inspection and packaging
S; D 0
However, because we shall be carrying out a variety of mathematical operations
using these data it will be useful to show the formulation in decimal notation rather
than fractional, giving:

Max 10S þ 9D
s:t:
0:7S þ 1D 630
0:5S þ 0:8333D 600
1S þ 0:6667D 708
0:1S þ 0:25D 135
S; D 0 (2:6)

Our job now is to find the product mix (i.e., the combination of S and D) that
satisfies all the constraints and, at the same time, yields the maximum possible value
for the objective function. Once these values are calculated, we will have found the
optimal solution to the problem.
This mathematical model of the problem is a linear programming model,or
linear programme. The problem has the objective and constraints that, as we said
earlier, are common properties of all linear programmes. But what is the special
feature of this mathematical model that makes it a linear programme? The special
feature that makes it a linear programme is that the objective function and all
constraint functions (the left-hand sides of the constraint inequalities) are linear
functions of the decision variables.
Mathematical functions in which each variable appears in a separate term and
Try Problem 1 to test is raised to the first power are called linear functions. The objective function
your ability to recognize (10S +9D) is linear because each decision variable appears in a separate term
the types of and has an exponent of 1. The amount of production time required in the
mathematical
relationships that can be cutting and dyeing department (0.7S +1D) is also a linear function of the
found in a linear decision variables for the same reason. Similarly, the functions on the left-hand
programme. side of all the constraint inequalities (the constraint functions) are linear func-
tions. Thus, the mathematical formulation of this problem is referred to as a
linear programme.
Linear programming has nothing to do with computer programming. The use of
the word programming here means ‘choosing a course of action’. Linear program-
ming involves choosing a course of action when the mathematical model of the
problem contains only linear functions.

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